Measuring and analyzing residual stresses and their gradients in materials using high resolution grazing incidence x-ray diffraction

ABSTRACT

A high resolution grazing incidence X-ray diffraction technique for measuring residual stresses and their gradients as a function of depth in thin film materials on substrates or in bulk materials is disclosed. The technique includes positioning a material relative to an X-ray source and an X-ray detector, performing an Omega scan to determine an Omega offset, setting the incidence angle at a first target incidence angle based on the Omega offset and greater than the critical angle of the material, performing a grazing incidence X-ray diffraction scan, analyzing the results to identify diffraction peaks, selecting a diffraction peak, setting the incidence angle at a second target incidence angle based on the Omega offset and a desired penetration depth, performing two theta scanning on a range of two theta values around the selected diffraction peak, performing refraction correction, and determining residual stress values for the material.

BACKGROUND

The present disclosure relates to thin film materials deposited, grown,or produced in any other manner on substrates or to bulk materials andthe measurement and analysis of the bulk materials or thin filmmaterials for residual stresses and their gradients using highresolution grazing incidence X-ray diffraction.

Stress may be defined as the ratio of force applied to a cross sectionalarea of a material. Stresses on a material may be either tensile orcompressive. Materials undergo elastic and plastic deformations as forceis applied. Plastic deformation is permanent non-recoverabledeformation. Elastic deformation is recoverable and temporary, e.g., nopermanent change is made to the material and the atoms return to theoriginal position when the force is removed.

Residual stresses are defined as elastic stresses that remain in thematerial following processing. Excessive residual stresses present inbulk materials or in thin film materials could lead to failure of thematerials during use. Residual elastic stresses in thin film materialdeposits and coatings may occur in many industries where thin filmmaterials and coatings are used and produced such as aerospace,automotive, biomedical, ceramics, coatings, electronics, energy, metals,optics, thin film material deposition tool manufacturing, Semiconductorand Packaging, and other similar industries. X-ray diffractiontechniques may be used to measure residual stresses in materials bymeasuring changes in the spacing between atomic planes in the materials,also known as d_(spacing) or d-spacing.

BRIEF SUMMARY

The system, method, and computer program product described hereinprovide ways of measuring and analyzing residual stresses and theirgradients in thin film materials deposited, grown, or produced in anyother manner on substrates or in bulk materials using high resolutiongrazing incidence X-ray diffraction.

In an aspect of the present disclosure, a method is disclosed thatincludes performing an Omega scan to determine an Omega offset of anX-ray beam generated by an X-ray source relative to a material withrespect to an incidence angle between the X-ray source and the material,setting the incidence angle between the X-ray source and the material ata first target incidence angle that is based on the Omega offset andgreater than the critical angle of the material, performing a grazingincidence X-ray diffraction scan on the material to generate firstmeasurement data including intensities of X-ray photons at a pluralityof two theta angles, analyzing the first measurement data to identify aplurality of diffraction peaks from the material each of which has anintensity occurring at a corresponding two theta value, selecting adiffraction peak of the plurality of diffraction peaks based on theanalysis of the first measurement data, setting the incidence anglebetween the X-ray source and the material at a second target incidenceangle based on the Omega offset and a desired penetration depth into thematerial, and performing two theta scanning on the material on a rangeof two theta values around the two theta value of the selecteddiffraction peak at a plurality of tilt positions to generate secondmeasurement data, applying refraction correction to the secondmeasurement data, the refraction correction correcting the secondmeasurement data for each tilt position of the scanned range of twotheta values around the two theta value of the selected diffractionpeak, converting the corrected second measurement data measured at eachtilt position to a d-spacing for each tilt position, and determiningresidual stress values of the material based on the converted correctedsecond measurement data.

In aspects of the present disclosure, apparatus, systems, and computerprogram products in accordance with the above aspect may also beprovided. Any of the above aspects may be combined without departingfrom the scope of the present disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The details of the present disclosure, both as to its structure andoperation, can be understood by referring to the accompanying drawings,in which like reference numbers and designations refer to like elements.

FIG. 1 is a diagram illustrating an example of a material undergoingcompressive stress.

FIG. 2 is a diagram illustrating an example of a material undergoingtensile stress.

FIG. 3 is a plot of intensity vs. Two Theta angle showing an overlap ofdiffraction peaks between a Nickel thin film material and a substrate ofa sample when a Two Theta-Omega scan (non-grazing type scan) is used.This scan type is utilized in the traditional sin² ψ stress techniqueand has limited control over penetration depth of X-rays.

FIG. 4 is a plot of intensity vs. Two Theta angle showing onlydiffraction peaks of a Nickel thin film material as measured by agrazing incidence diffraction technique in accordance with some aspectsof the present disclosure.

FIG. 5a is an image of a testing apparatus in accordance with someaspects of the present disclosure.

FIG. 5b is another image of the testing apparatus of FIG. 5a inaccordance with some aspects of the present disclosure.

FIG. 6 is an image of an alternative testing apparatus in accordancewith some aspects of the present disclosure.

FIG. 7 is a flow chart of a method for calibrating the testing apparatususing a Theta-Two Theta (Horizontal Goniometer) or Theta-Theta (VerticalGoniometer) diffractometer (FIGS. 5a, 5b , and 6) in accordance withsome aspects of the present disclosure.

FIGS. 8-13 are diagrams illustrating some aspects of the calibrationmethod of FIG. 7 in accordance with some aspects of the presentdisclosure.

FIG. 14 is a plot showing an example of a recording of X-ray photonintensity coming from angular measurement of source/Omega/theta, asrecorded by a detector, during calibration in accordance with someaspects of the present disclosure.

FIG. 15 is an example of a sample in accordance with some aspects of thepresent disclosure.

FIG. 16 is a flow chart of a method for performing high resolutiongrazing incidence X-ray diffraction in accordance with some aspects ofthe present disclosure.

FIGS. 16a and 16b describe projection of the normal of stage/sample tilt(psi) on the sample surface.

FIGS. 17 and 18 are diagrams illustrating the positioning in accordancewith the method of FIG. 16.

FIG. 19 is a diagram illustrating the rocking of a sample to determinean Omega offset when using a Theta-Two Theta/Horizontal GoniometerDiffractometer according to a portion of the method of FIG. 16.

FIG. 19a is a diagram illustrating the effects of improper samplemounting for a Theta-Two Theta /Horizontal Goniometer Diffractometer.

FIG. 19b is a figurative description of the penetration depth of X-raysinto a thin film material or a bulk material.

FIG. 20 is a flow chart of a method for analyzing measurement data inaccordance with some aspects of the present disclosure.

FIG. 21 is a diagram illustrating the effects of refraction on a samplein accordance with some aspects of the present disclosure.

FIG. 22 is a plot of strain vs. cos² alpha*sin² ψ based on measurementsfrom a Tungsten thin film material sample using the high resolutiongrazing incidence X-ray diffraction technique in accordance with someaspects of the present disclosure.

FIG. 23 is a plot of strain vs. sin² ψ based on measurements from aTungsten thin film material sample using the traditional sin² ψ inaccordance with some aspects of the present disclosure.

FIG. 24 is a block diagram of a computing device that may be used forperforming some or all of the high resolution grazing incidence X-raydiffraction technique in accordance with some aspects of the presentdisclosure.

FIG. 25 is an exemplary block diagram of a computer system in whichprocesses involved in the system, method, and computer program productdescribed herein may be implemented.

DETAILED DESCRIPTION

With reference now to FIG. 1, an example of a polycrystalline material100 experiencing a compressive residual stress is illustrated. Residualstresses are defined as stresses that remain in the material followingprocessing. They are elastic in nature and do not result in anypermanent deformation/change to the material. In the case of compressiveresidual stresses, for example, as seen in FIG. 1, the spacing betweenatomic planes of crystallites, in the direction of application of force,as indicated in 102 is decreased and at the same time the verticalspacing between the atomic planes is increased. The spacing betweenatomic planes is also denoted as d_(spacing). With reference now to FIG.2, material 100 experiencing a tensile residual stress is illustrated.In the case of tensile residual stresses, for example, as seen in FIG.2, the spacing between atomic planes of crystallites, in the directionof application of force, as indicated in 102 is increased and at thesame time the vertical spacing between the atomic planes is decreased.

Residual stresses may be measured in a non-destructive fashion usingX-ray diffraction. One common technique is the traditional sin² ψtechnique. The changes in the spacing between atomic planes(d_(spacing)) of a high angle (e.g., Two-Theta) reflection may beplotted against sin² ψ, where, ψ describes the angular motion of thesample (sample tilt). The radius of the circle formed by the ψ motion isperpendicular to the radius of circle created by the angular motion ofthe source and detector. This ψ motion is also shown in FIG. 5A, wherethe sample stays flat when ψ equals 0 degrees, and in FIG. 6 the samesample is at a tilted position and in this case ψ equals 45 degrees. Theresidual stress may be calculated from the slope of the plot usingelastic constants (e.g., Young's Modulus and Poisson's ratio). Thetraditional sin² ψ technique may be applied to measure residual stressesin bulk materials or in the case of thin film materials/coatings wherethere is no overlap between reflections or diffraction peaks from thethin film material and substrate or other underlying layers. However thetraditional sin² ψ technique incapable of precisely measuring residualstresses present near the surface or subsurface regions in bulkmaterials. In the case of thin film materials/coatings, the traditionalsin 2 ψ technique may be incapable of precisely measuring residualstresses in the thin film material when there is an overlap betweenreflections or diffraction peaks of the thin film material and thesubstrate. For example, the sin² ψ technique completely fails whenanalyzing residual stresses in the case of metallic thin film materialsdeposited on carbon based substrates, e.g., nickel deposited on graphiteor other similar materials, due to extensive interference betweenreflections from both the Nickel thin film material and the carbon basedsubstrate as shown, for example, in FIG. 3. As seen in FIG. 3, forexample, the nickel and graphite diffraction peaks occur atapproximately the same Two Theta values. Because of this, it may be hardto separate the reflections or signals received from the Nickel thinfilm material from the reflections or signals received from thesubstrate using the traditional sin² ψ technique and therefore todetermine the residual stresses of the Nickel thin film material.

One approach to avoid reflections or signals from the substrate, orunderneath layers in the case of multiple stack films, may include theuse of a Grazing Incidence Diffraction technique. In the GrazingIncidence Diffraction technique, the X-ray beam may be applied to thesample at a shallow angle with respect to the sample surface. Due to theshallow angle, the X-ray beam may not penetrate deeply into the sampleand instead the X-ray beam may only reflect from the top layer of thesample to be captured by the detector. At higher grazing incidentangles, however, significant peak overlap between reflections from thesubstrate and the film may be present and may require the use ofdedicated hardware to minimize overlap. An example of a GrazingIncidence Diffraction result on the nickel coated graphite system isillustrated in FIG. 4. As can be seen in FIG. 4, for example, only thediffraction peaks associated with the nickel coating are present in theDiffractogram.

Disclosed herein is a high resolution grazing incidence X-raydiffraction technique for measuring residual stresses and theirgradients as a function of depth in thin films/coatings (e.g., be ametal, polymer, ceramic, oxide, organic, or other thin films/coatings)that has been deposited, grown naturally, or produced in any othermanner on a substrate/carrier. This technique may be applied to bulkmaterials as well. In the case of bulk materials this technique mayprovide some advantages for quantifying surface residual stresses nearthe surface of the bulk material or near boundaries of the bulk materiallayers that may be introduced by processes such as shot peening,cutting, surface polishing, etching, ion implantation, etc. As usedherein, the term material may encompass both thin film materialsdeposited, grown, or produced in any other manner on substrates and bulkmaterials.

Testing Apparatus

With reference now to FIGS. 5a and 5b , an example testing apparatus 500for measuring and analyzing residual stresses using the High ResolutionGrazing Incidence X-ray Diffraction technique is illustrated. Testingapparatus 500 includes a source 502, a detector 504, and a stage 506.Testing apparatus 500 may utilize a Theta-Theta or vertical goniometerconfiguration. In this configuration source 502 and detector 504 may beconfigured to move in an angular fashion relative to center point 508(FIG. 8) of testing apparatus 500. For example, an angle between source502 and stage 506 or center point 508 (FIG. 8) may be created by movingthe source in an angular fashion. Likewise, an angle between detector504 and stage 506 or center point 508 (FIG. 8) may be created by movingthe detector in an angular fashion. In some aspects, source 502 anddetector 504 may physically translate, rotate, or move to adjust theangle between source 502 and stage 506 or center point 508 (FIG. 8) andbetween detector 504 and stage 506 or center point 508 (FIG. 8). In someaspects, stage 506 may be fixed in position, orientation, or both. Insome aspects, stage 506 may be physically translated, rotated, or movedrelative to source 502 and detector 504, as illustrated, for example, inFIG. 5b . In some aspects, for example, one or both of source 502 anddetector 504 may be fixed in position, orientation, or both relative tothe location of stage 506 or center point 508 (FIG. 8).

In some aspects, testing apparatus 500 may be or include a goniometer,an X-ray diffractometer, or both. For example, testing apparatus 500 maybe configured to precisely measure the angles between source 502,detector 504, stage 506, and center point 508 during X-ray testing. Insome aspects, testing apparatus 500 may include an X-ray diffractometerwith a rotating anode, microfocus rotating anode, a sealed tube,microfocus X-ray source, or other similar hardware.

Another form of test apparatus that may utilized for the High ResolutionGrazing Incidence X-ray Diffraction technique is a Theta-Two Theta orHorizontal Goniometer diffractometer, as illustrated, for example, inFIG. 6. Testing apparatus 600 includes a source 602, a detector 604, anda stage 606. In this case, for example, as illustrated in FIG. 19, anangle between source 602 and stage 606 or a center point 508 (FIG. 8) oftesting apparatus 600 may be created by moving the stage in an angularfashion. Likewise, an angle between detector 604 and stage 606 or centerpoint 508 (FIG. 8) may be created by moving the detector in an angularfashion. In some aspects, source 602 and detector 604 may physicallytranslate, rotate, or move to adjust the angle between source 602 andstage 606 or center point 508 (FIG. 8) and between detector 604 andstage 606 or center point 508 (FIG. 8). In some aspects, stage 606 maybe fixed in position, orientation, or both. In some aspects, stage 606may be physically translated, rotated, or moved relative to source 602and detector 604.

Another form of test apparatus that may be utilized for the HighResolution Grazing Incidence X-ray Diffraction technique is a Theta-TwoTheta configuration diffractometer using a Vertical Goniometer.

Another form of test apparatus that may be utilized for the HighResolution Grazing Incidence X-ray Diffraction technique is andiffractometer where the stage can be moved up or down vertically in acontrolled fashion in order to maintain a constant or variable distancebetween sample and detector.

Another form of test apparatus that may be utilized for the HighResolution Grazing Incidence X-ray Diffraction technique is a grazingincidence bench top style compact diffractometer where the source isfixed at a shallow incidence angle or has limited motion. The detectormay move in an angular fashion or may remain fixed or may haverestricted motion.

Another form of test apparatus that may be utilized for the HighResolution Grazing Incidence X-ray Diffraction technique is adiffractometer where multiple or single X-ray producing sources and/ordetectors are used.

While described herein with reference to source 502, detector 504, andstage 506 of testing apparatus 500, any of the above mentioned testingapparatus may be utilized to perform the High Resolution GrazingIncidence X-ray Diffraction technique.

Source 502 may be any device that is configured to produce or supplyelectromagnetic radiation or beam 510 (FIG. 8) toward stage 506 anddetector 504. For example, source 502 may be configured to produce orsupply an X-ray beam. An example source 502 may include, for example, aSealed X-ray Tube, Rotating anode, microfocus X-ray tube, microfocusrotating anode, or other similar X-ray sources, utilizing single ormultiple solid targets (liquid or solid or both) to produce X-rays ofdesired wavelength/wavelengths.

Detector 502 may be any device that is configured to sense an intensityof electromagnetic radiation, e.g., beam 510 (FIG. 8), received eitherdirectly from source 502 or indirectly from source 502 via a samplepositioned or attached to stage 506, e.g., due to reflection,refraction, diffraction, or other similar radiation from the sample. Insome aspects, for example, detector 502 may be gas based or ascintillator or silicon strip/semiconductor based or may use any othertechnology which is configured to measure the intensity of an X-raybeam. In some aspects, detector 504 may be a 2D, 1D, or 0D X-raydetector. In some aspects, detector 504 may be a combined 1D/0Ddetector. In some aspects, detector 504 may be a combined 2D/1D/0Ddetector. The detectors may or may not have energy discriminationcapability. An example detector 504 may include, e.g. Scintillationdetectors (e.g., offered by Bruker AXS, Panalytical, Rigaku or othervendors), Pacel® (offered by Panalytical), X'Celerator® (offered byPanalytical), Vantec™ (offered by Bruker AXS), LynxEye™/LynxEye XE™(offered by Bruker AXS), D/Tex™ (offered by Rigaku), Hypix® 3000(offered by Rigaku), etc.

In some aspects, the different kinds of testing apparatus describedabove may include additional components, for example, source side orprimary optics. Some of the components may include an X-ray mirror ormirrors, polycapillary optics, or collimators that provides an X-raybeam that is either parallel, pseudo parallel, or focusing in point/Lineform. In some aspects, the testing apparatus may include multiple orsingle mirror exit slits, masks, pinholes, or collimators that may beutilized to control the width, height, and shape of the X-ray beamhitting the sample. For example, the mirror exit slit may be a 0.2 mmslit. As another example, the mirror exit slit may be a 0.1 mm pinholeslit. In some aspects, the mirror exit slit may be changed or swappedout, e.g., exchanging the 1 mm slit for the 0.1 mm slit, to control thespread of the X-ray beam on the sample and determine what portion of thesample should contribute to the stress analysis.

In some aspects, the testing apparatus may include a cradle havingmultiple axes of freedom such as, e.g., a Chi axis of freedom (sampletilt), Phi axis of freedom (sample rotation), or other axes of freedom.

In some aspects, the testing apparatus may include a Hexapod stage orits variants where parallel-kinematic micro positioning systems are usedto create axes of freedom such as, e.g., a Chi axis of freedom (sampletilt), Phi axis of freedom (sample rotation), or other axes of freedom.

In some aspects, testing apparatus may include use of axialsoller/sollers that may be used to control the divergence of the X-raybeam in the axial direction, e.g., perpendicular to the direction ofpropagation of the X-ray beam in the plane of the paper. In someaspects, for example, the axial soller may range from 0.05 degrees to afew degrees.

In some aspects, testing apparatus may include an equatorial soller thatdefines the Two Theta of the diffracted beam precisely and separates outreflections from thin film material and substrate. In some aspects, theequatorial soller may range from 0.05 degrees to a few degrees, e.g., 1,2, 3, or more degrees or any other degree value. In some aspects, across soller may be utilized, as it acts as an equatorial as well as anaxial soller.

In some aspects, testing apparatus may include a scintillation detector.In some aspects the testing apparatus may include either a 2D, 1D, or 0DX-ray detector. In some aspects the testing apparatus may includecombined 1D/0D detector. In some aspects the testing apparatus mayinclude a combined 0D/1D/2D detector.

Apparatus Calibration

With reference now to FIGS. 5a, 5b , and 6-14, testing apparatus may becalibrated or aligned prior to testing of a sample. In some aspects, forexample, testing apparatus may be calibrated using common industrypractices. In some aspects, for example, testing apparatus may becalibrated using inputs from a vendor. In FIGS. 8-10, X, Y, and Z axesare illustrated with the Z axis extending out of the page perpendicularto the X and Y axes. In FIGS. 11-13, X, Y, and Z axes are illustratedwith the X axis extending out of the page perpendicular to the Y and Zaxes.

At 702, prior to positioning a sample on stage 506, the center point 508of testing apparatus 500 may be determined. In some aspects, forexample, the center point 508 may be determined by positioning analignment fixture (not shown) on stage 506 and camera/scope/videocamera/visual device to observe motion of fixture, a laser, or any othermethod provided by a manufacturer of the stage 506 or other componentsof testing apparatus. In addition a level or a similar device may beused to ensure that the stage is perfectly flat at ψ=0 degrees. In someaspects, the Chi (sample tilt) and the Phi (sample rotation) axes of thestage should exhibit little or no wobble and should be withinmanufacturer specifications.

At 704, source 502 and/or primary optics may be calibrated to ensurethat a parallel/pseudo parallel beam 510 with a desired wavelength inthe form of a spot or line beam. For example, the source 502 and/orprimary optics may be configured to produce a focusing X-ray beam havinga desired X-ray form, shape, focus, wavelength, intensity, or othersimilar parameters. In some aspects, the desired wavelength may bedetermined based on the sample composition. For example, in the case ofresidual stress measurements on Copper films, Copper K alpha radiationmay be used. However the same radiation type may not be ideally suitedfor analyzing residual stresses in Cobalt films due to fluorescenceeffects since the Copper radiation causes the sample to act as asecondary source of X-rays, thereby drastically lowering the peak tobackground for the Cobalt diffraction peaks.

At 706, a slit 512 may be placed in front of detector 504. For example,a 0.05 mm slit 512 may be placed in front of detector 504. The slitsizes can range from 0.01 mm to a few mm. The slit may be configured toreceive the x-ray beam.

At 708, the source and detector alignment may be checked to determinewhether they are in or approximately in a straight line, e.g., notpositioned at an angle with respect to each other, and parallel orapproximately parallel to the ground. This may occur when source anddetector are positioned close to zero degree positions as shown, forexample, in FIG. 8. A detector scan may be performed by moving orrocking the detector 504 a few degrees (for example, −5 to 5 degrees)relative to the zero position, e.g., angular motion. In some aspects,nothing should be placed between source 502 and detector 504. During thedetector scan, detector 504 may detect changes in intensity values fromthe received beam 510 by moving between −5 and 5 degrees.

At 710, a Gaussian center of the detected changes in intensity valuesfrom the received beam 510 during the detector scan may be defined orre-defined as a zero position for the axes of detector 504. The detector504 may be calibrated based on the defined zero position such that a 0degree marking for the detector 504 is the defined zero position.

At 712, an alignment fixture 514, e.g., as illustrated in FIGS. 9, 10,and 12 and 13, may be positioned at the center point 508 of the testingapparatus 500, e.g., on the stage 506. Alignment fixture 514 may includea pathway 516, e.g., a slot, channel, hole, or other similar feature,that extends through alignment fixture 514 and is configured to receivethe beam 510 there through when properly aligned.

At 713, a slit in front of the detector may be removed or the detectorslit size may be limited, e.g., to 3 mm or another value.

At 714, the source and/or primary optics may also need to be translateda few mm relative to center point to check and ensure if the beam isgoing through the center point. A fluorescent screen jig or similarfixture (not shown) may be used for this part of the procedure.Following translation, with reference to FIG. 8-10, since source 502 maybe fixed, the alignment fixture 514 needs to be rocked, e.g., rotatedabout the center point 508 between −5 and 5 degrees or other similarvalues, relative to source 502 while source 502 supplies beam 510 towarddetector 504. In some aspects, with reference to FIG. 11-13, forexample, the stage 506 and alignment fixture 514 may be fixed and source502 may instead be physically rocked or rotated, e.g., between −5 and 5degrees. These scans are termed as Theta-Theta Omega/source scans orTheta-Two Theta rocking curve scans, depending on the set up of theapparatus. As seen in FIGS. 9, 10, 12, and 13, during rocking, the beam510 only reaches detector 504 when the pathway 516 is aligned orpartially aligned with the source 502 and detector 504 such that some orall of beam 510 travels through the pathway 516.

With reference now to FIG. 14, an example output measurement by detector504 from the rocking of alignment fixture is illustrated. As seen inFIG. 14, as the alignment fixture is rocked between −0.5 and 0.5degrees, an intensity of the beam 510 forms a bell shaped curve due tothe alignment fixture 514 blocking or inhibiting some or all of the beam510 from reaching detector 504. For example, the highest detectedintensity level may correspond to the position of alignment fixturewhere pathway 516 is fully aligned between source 502 and detector 504such that the beam 510 may pass directly through pathway 516 to reachdetector 504

At 716, referring to FIGS. 7 and 14, the Gaussian center of the detectoroutput illustrated in FIG. 14 may be determined and the Theta axes(created by either stage or source motion) may be calibrated to set thedetermined Gaussian center to zero degrees to ensure that the beam 510goes through the center point 508.

At 718, slit 512 may once again be inserted in front of detector 504 anda detector scan may be run to re-determine the zero position of thedetector 504 in the manner described above at 706-710.

In some aspects, following step 718, Omega scans or rocking curvemeasurements may be carried out a pre-determined number of times todetermine precision in the source or stage motion respectively. In someaspects, a desired precision may be a precision better than 0.005degrees. For example, in some aspects, 200-300 times or more Omega scansmay be generated, in order to create the necessary set of data pointsfor statistical error analysis. Any other number of times may be used.In some aspects, similar measurements may be carried out for thedetector axes, using procedure described in step 718, in order todetermine its precision.

Testing apparatus 500 is now calibrated for testing. In some aspects,testing apparatus 500 may be calibrated periodically, e.g., annually,quarterly, monthly, weekly, daily, before testing a batch of samples, orbefore the testing each sample.

High Resolution Grazing Incidence X-ray Diffraction Technique

Once the testing apparatus 500 is calibrated, the high resolutiongrazing incidence X-ray diffraction technique may be performed on asample 518 with reference now to FIGS. 15-23. In FIGS. 17-19, X, Y, andZ axes are illustrated with the Z axis extending out of the page towardsthe reader and is perpendicular to the X and Y axes.

FIG. 15 illustrates an example sample 518 to be tested using the highresolution grazing incidence X-ray diffraction technique, in this case,a nickel coating on a graphite substrate is chosen. Sample 518 could beany thin film material/coating 520 (e.g., metal, polymer, ceramic,oxide, organic, etc.) that has been deposited, grown naturally, orproduced in any other manner on the substrate 522. For example, thecoating or thin film material 520 may be deposited on substrate 522using, e.g., Thin Film Deposition or Electrodeposition or otherwell-known techniques. In some aspects, a thin film material may, forexample, be a layer of material ranging from a few nanometers to severalmicrons in thickness. Thin film material 520 includes a top surface 524and a bottom surface 526 Other samples including the same or differentthin film materials deposited on the same or different substrates mayalso be tested using the high resolution grazing incidence X-raydiffraction technique. In some aspects, sample 518 may alternativelyinclude a bulk material. While described generally with reference tothin film material 520, the technique described herein may be utilizedon both to samples including bulk materials and samples including thinfilm materials.

With reference now to FIG. 16, a method 1600 for performing highresolution grazing incidence X-ray diffraction on sample 518 including amaterial, e.g., a thin film deposited, grown, or produced in any othermanner on a substrate or a bulk material, will now be disclosed.

Sample Positioning and Alignment

In some aspects, the sample may be positioned on the stage such that thedirection of measurement of stress is parallel to the projection of thenormal of stage/sample tilt (psi) on the sample surface, as shown, forexample, in FIGS. 16a and 16b . FIG. 16a illustrates a top view of thesample showing the direction of the incident x-rays and the direction ofthe normal y which shows that the residual stress is measured along they direction for a Theta-Theta configuration. FIG. 16b illustrates a sideview of the sample, showing the sample stage normal at various psivalues and the normal y. In some aspects, the desired direction ofmeasurement may be adjusted by sample rotation or by manual means. Insome instances stress measurements may be carried out along multipledirections on the same sample. In some aspects, a deviation from theabove positioning of the sample may result in reduced accuracy ofmeasurement or invalid results.

Alignment of sample 518 may affect the results of the High ResolutionGrazing Incidence X-ray Diffraction measurements as the sample heightand flatness of sample surface with respect to the Incident X-ray beammay impact the measurement accuracy and results.

At 1602, sample 518 may be positioned and aligned on stage 506 asillustrated, for example, in FIG. 17, and as described above. At 1604,stage 506 may be actuated or adjusted to position the sample 518 inbetween source 502 and detector 504 as illustrated, for example, in FIG.18, where the source 502 and detector 504 are at zero degrees. In someaspects, sample 518 may be moved between source 502 and detector 504such that sample 518 is positioned approximately halfway into beam 510,as illustrated, for example, in FIG. 18. For example, before the sample518 is positioned between source 502 and detector 504, a photonintensity of the beam 510 as measured by the detector 504 may be, e.g.,100 k. As the sample 518 is moved into position between source 502 anddetector 504, the photon intensity of the beam 510 as measured bydetector 504 may be reduced. For example, when sample 518 is positionedapproximately halfway into the beam 510, the photon intensity of thebeam 510 as measured by the detector 504 may be reduced to, e.g., 50 k.This procedure may also be termed as sample height adjustment or Z scan.

In some aspects, the sample may be aligned by performing a Z scan (e.g.,the sample height is adjusted to half of the intensity of the PrimaryX-ray beam), followed by a ψ scan from −5 to 5 degrees, following whichthe sample is moved to the Gaussian mid-point on the ψ scan. This helpsto correct for sample tilts in the orthogonal direction to the X-raybeam 510. This is followed by a Z scan, rocking curve scan at Two Thetaequals zero (e.g., the sample is either rocked in the Omega direction byrotating stage about center point or the source and detector are movedin order to carry create the rocking curve motion, depending on the testapparatus geometry), following which the stage or source (depending onthe test apparatus) is moved to the Gaussian mid-point on the rockingcurve scan. This is followed by another Z scan. If the mid-point valuesof Z scan (before and after rocking curve) differ by, e.g., 20%, thealignment procedure may be repeated at least one or more times. Otherpercentages of difference may also trigger a repeat of the alignmentprocedure.

In some aspects, the sample may be aligned by performing a Z scan (e.g.,the sample height is adjusted to half of the intensity of the PrimaryX-ray beam), rocking curve scan at Two Theta equals zero (e.g., thesample is either rocked in the Omega direction by rotating stage aboutcenter point or the source and detector are moved in order to carrycreate the rocking curve motion, depending on the test apparatusgeometry), following which the stage or source (depending on the testapparatus) is moved to the Gaussian mid-point on the rocking curve scan.This is followed by another Z scan. If the mid-point values of Z scan(before and after rocking curve) differ by 20%, the alignment proceduremay be repeated at least one or more times. Other percentages ofdifference may also trigger a repeat of the alignment procedure.

In some aspects, the sample may be aligned by using manual and/ormotorized adjustment of two orthogonal tilt angles. This may be presentas an option on some stages or may be available as on add on option toexisting stage. The sample may be aligned by performing a Z scan (e.g.,the sample height is adjusted to half of the intensity of the PrimaryX-ray beam), followed by rocking curve scan using one of the orthogonaltilt axes (only the orthogonal tilt axis is moving in this case, sourceis fixed and detector is fixed). Followed by sample rotation (phi) to 90degrees, where another orthogonal axis scan is carried out. This isfollowed by another Z scan. If the mid-point values of the orthogonal Zscans differ by 20%, the alignment procedure may be repeated at leastone or more times. Other percentages of difference may also trigger arepeat of the alignment procedure.

Determining the Omega Offset

At 1606, the Omega offset of the sample 518 is determined. The Omegaoffset is the angle from 0 degrees that the source must be adjusted forthe sample 518 due to unintended sample tilt brought upon by non-uniformsample mounting or non-uniformity in stage machining or a combination ofother factors, as shown, for example, in FIG. 19 a.

The Omega offset may be determined by performing a rocking curve scan atTwo Theta equals zero (e.g., the sample is either rocked in the Omegadirection by rotating stage about center point or the source anddetector are moved in order to carry create the rocking curve motion,depending on the test apparatus geometry), following which the stage orsource (depending on the test apparatus) is moved to the Gaussianmid-point on the rocking curve scan. During the Omega scan, detector 504measures the intensity of the beam 510 as it reflects off of sample 518.If the sample 520 were to be perfectly flat, the measurement by detector504 will show a Gaussian center at 0 degrees for Omega. In this case,the Omega offset is 0. If, however, the sample was not flat, theGaussian center of the intensity measurements by detector 504 may not beat 0 degrees for Omega. In this case, the Omega offset is determined tobe the offset between 0 degrees and the Gaussian center of the intensitymeasurements. For example, if the Gaussian center of the intensitymeasurements is at 0.08 degrees, an Omega offset of 0.08 degrees may bedetermined. The determined Omega offset may be stored for later use,e.g., in memory of a computing device.

If the Omega value obtained during alignment is non-zero, theexperimental Omega value used during the actual measurement needs to bechanged. For example, if during alignment the Omega position isdetermined to be located around 0.08 degrees and the user wants toperform a high resolution grazing incidence X-ray diffraction scan at0.5 degrees. The Omega drive for the measurement should be set to 0.5degrees+0.08 degrees, e.g., 0.58 degrees, and not 0.5 degrees, toaccount for the Omega offset. In some instances the Omega offset may beas high as 1.5 degrees or more. In this case the Omega drive for themeasurement should be set to 0.5 degrees+1.5 degrees, e.g., 2 degrees,and not 0.5 degrees, to account for the Omega offset. In some instancesthe offset may be −0.08 degrees or lower. In this case the Omega drivefor the measurement should be set to 0.5 degrees+(−0.08) degrees, e.g.,0.42 degrees, and not 0.5 degrees, to account for the Omega offset. Thisadjustment may account for any sample tilts and sets the incident angleof the X-ray beam 510 accurately with respect to the sample surface.This adjustment ensures that the X-ray beam is penetrating into thesample at the desired incident angle used during the High resolutionGrazing Incidence stress measurement.

Reposition Sample

At 1608, once the Omega scan is complete, the sample 518 may berepositioned such that the angle of sample 518 relative to the source502 is equal to the target scan incident angle plus the Omega offset.The first target incidence angle is chosen such that it is slightlygreater than the critical angle of the bulk material or thin filmmaterial. In some aspects, for example, slightly greater may be 0.01degrees greater, 0.02 degrees greater, or other similarly small angles.For example, if the critical angle of the bulk material or thin filmmaterial is 0.48 degrees, the first target incidence angle may be 0.5degrees. The critical angle may be determined by performing X-rayreflectometry measurements on the sample (e.g., using the testingapparatus described herein) or may be determined from theoreticalcalculation described below.

critical angle=π(2*δ)   (1)

The susceptibility δ of the material may be determined according to thefollowing equation:

$\begin{matrix}\left. {\delta = {2.78 \times 10^{8}{\frac{p\; \lambda}{M}\left\lbrack {Z + {\Delta \; f^{\prime}}} \right)}}} \right\rbrack & (2)\end{matrix}$

Where:

p is the density of the material;

Z is the atomic number of the material;

Δf′ is the real part of dispersion correction of scattering factor forthe material;

A is the wavelength of the X-rays; and

M is the atomic mass of the material.

Identify Diffraction Peaks From Bulk Material or Thin Film Material

At 1610, once the sample 518 has been calibrated, the source 502 may beset at a pre-determined first target incident angle to perform a grazingincidence diffraction scan on the sample. As mentioned above, thepre-determined incident angle is adjusted by the Omega offset prior toscanning. The grazing incidence diffraction may be used to identifydiffraction peaks from a thin film material deposited, grown, orproduced in any other manner on a substrate to selectively obtaindiffraction peaks from the thin film material without interference fromthe substrate. In the case of bulk materials, the grazing incidencediffraction technique can be used to selectively identify diffractionpeaks from surface or sub surface regions of the bulk material. Duringthe grazing incidence diffraction, the angle between source 502 andsample 518 may be fixed while the angle of detector 504 may beadjusted/moved through a range of angles to generate intensitymeasurements. For example, as illustrated in FIG. 4, the intensity ofdiffracted X-ray photons may be plotted against the Two Theta angle todetermine which diffraction peaks having intensity have occurred. Asillustrated in FIG. 4, for example, five diffraction peaks/reflectionshaving sufficient intensity are present for the thin film material,e.g., Nickel, peak 1 at approximately 44.51 degrees (Ni, (1 1 1)), peak2 at approximately 51.85 degrees (Ni, (2 0 0)), peak 3 at approximately76.37 degrees (Ni, (2 2 0)), peak 4 at approximately 92.94 degrees (Ni,(3 1 1)), and peak 5 at approximately 98.45 degrees (Ni, (2 2 2)). Byperforming the grazing incidence diffraction scan, the diffraction peaksfrom the thin film material may be determined without interference fromthe substrate Likewise, diffraction peaks from a surface region or subsurface region of the bulk material may be determined withoutinterference from other regions of the bulk material.

Select Desired Diffraction Peak/Reflection for Further Scanning

At 1612, once the locations of the diffraction peaks from the bulkmaterial or thin film material have been identified, further scanningmay be performed. In some aspects, for example, one of the diffractionpeaks may be selected for further scanning. In some aspects, thediffraction peak having highest intensity may be selected. For example,peak Ni (1 1 1) of FIG. 4 may be selected as the highest intensitydiffraction peak.

In some aspects, for example, the diffraction peak that has the largestspacing in Two Theta degrees from adjacent peaks may be selected. Forexample, peak Ni (1 1 1) from FIG. 4, has a Two Theta angle ofapproximately 44.51 degrees. The closest adjacent peak is Ni (2 0 0)which has a Two Theta angle of approximately 51.85 degrees. Thus thespacing between peak Ni (1 1 1) and Ni (2 0 0) is approximately 7.34degrees. As another example, peak Ni (2 2 0) has a Two Theta angle ofapproximately 76.37 degrees. The adjacent peaks are Ni (3 1 1), whichhas a Two Theta angle of approximately 92.94 degrees, and Ni (2 0 0)which has a Two Theta angle of approximately 51.85 degrees. The spacingbetween peak Ni (2 2 0) and peak Ni (3 1 1) is approximately 16.57degrees and the spacing between peak Ni (2 2 0) and peak Ni (2 0 0) isapproximately 24.52 degrees. Thus peak Ni (2 2 0) has a larger spacingfrom adjacent peaks than peak Ni (1 1 1). Selecting the peak with thelargest spacing in Two Theta degrees from adjacent peaks may allow theselected peak to be further scanned with minimum/no interference fromother peaks. For example, the intensity values surrounding peak Ni (2 20) are at a minimal value for a larger amount of Two Theta values beforethe intensity increases to the adjacent peaks Ni (2 0 0) and Ni (3 1 1).

Although example criteria for selecting a particular diffraction peak isdescribed above, any other method of selecting diffraction peaks may beemployed and further scanning may be performed on any other peak. Insome aspects, more than one peak may be selected for a further round ofscanning.

Setting the Second Target Incident Angle

At 1614, once a diffraction peak from the material of interest, e.g.,thin film material or bulk material, has been selected from the initialgrazing incidence diffraction measurements, a second target incidenceangle relative to the sample based on the Omega offset and desiredpenetration depth into the bulk material or thin film material isselected.

The penetration depth of X-rays at a fixed incident angle, as describedin FIG. 19b , can be estimated using the following equation:

$\begin{matrix}{{D_{\max}\mspace{14mu} {or}\mspace{14mu} X\text{-}{ray}\mspace{14mu} {penetration}\mspace{14mu} {depth}} = \frac{4.606^{*}\sin \mspace{14mu} \left( {{incident}\mspace{14mu} {angle}} \right)}{2*\left( {u/p} \right)^{*}p}} & (3)\end{matrix}$

Where:

p is the density of the material;

u/p is the mass attenuation coefficient of X-rays for the material;

In some aspects, one or more incident angles other than second targetincidence angle may be chosen. For example, in the case of Nickel films,an incident angle of 0.5 degrees, an incident angle of 5 degrees, and anincident angle of 10 degrees may be used. In the case of Nickel, anincident angle of 0.5 degrees may achieve a penetration depth of 0.46μm, an incident angle of 5 degrees may achieve a penetration depth of4.25 μm, and an incident angle of 10 degrees may achieve a penetrationdepth of 7.77 μm.

Scanning of Selected Diffraction Peak at Different Sample Tilts (ψ)

Once the diffraction peak and incident angle have been selected, TwoTheta scans around the selected diffraction peak may be carried out atdifferent sample tilt/psi/ψ positions at 1616. However the incidentangle is at a fixed position. During the Two Theta scan, the diffractedX-ray photon intensity and Two Theta position is recorded by scanningthe detector around the Two Theta position of the selected diffractionpeak. The ψ values used may range from 0 to 90 degrees. In someinstances a limited ψ range may be used such as 0 to 45 degrees. In someinstances fixed or varied intervals may be used for a particular ψrange. In some instances two ψ values may be used, such as 0 and 90degrees. In other instances two ψ values may be used, such as 0 and 45degrees. In some instances two or more ψ values may be used, such as 0,10, 20, 30, 40, 50, or any other number of degrees.

Analyzing the Measurement Data

At 1618 the measurement data, e.g., the series of intensity of X-rayphotons vs. Two Theta collected at different sample tilt (psi)positions, may be analyzed to determine residual stress values at acertain depth into the sample, while taking into account the Two Thetapeak shift brought upon by refraction of X-rays at shallow incidentangles according to the method illustrated in FIG. 20.

At 2002 a variety of inputs may be used as a basis for analysis of themeasurement data. For example, the inputs may include incident anglesadjusted by Omega offset, the Two Theta peak angles, ψ positions, theMiller indices for the selected diffracted peak/peaks, Young's Modulusvalues for selected diffracted peak/peaks, Poisson's ratio, the atomicweight of one or both of the elements constituting the bulk material orthin film material, the density of one or more element constituting thebulk material or thin film material, a real part of dispersion of one ormore elements constituting the bulk material or thin film material, andan atomic number of one or more elements constituting the bulk materialor thin film material. Some or all of these inputs may be determined oridentified with reference to known sources on material properties orcomposition including, for example, periodic tables, materials charts,text books, or other similar sources.

Some inputs that may be used for calculating stress may include the TwoTheta peak positions (obtained from experimental data either by manualselection, using peak deconvolution using standard, user made, combineddeconvolution methods/algorithms, or using other methods or acombination of methods), ψ positions (used during experiment), correctedincidence angle used for measurement (which includes Omega Offset), andelastic constants (obtained from Material properties found in Hand booksor other reliable sources).

Determine Susceptibility of the Material

At 2004, the susceptibility δ of the material may be determinedaccording to the following equation:

$\begin{matrix}\left. {\delta = {2.78 \times 10^{8}{\frac{p\; \lambda}{M}\left\lbrack {Z + {\Delta \; f^{\prime}}} \right)}}} \right\rbrack & (4)\end{matrix}$

Where:

p is the density of the material;

Z is the atomic number of the material;

Δf′ is the real part of dispersion correction of scattering factor forthe material;

λ is the wavelength of the X-rays; and

M is the atomic mass of the material.

Determining Refractive Index Shift

At 2006, the two theta shift due to refraction effects may becalculated, e.g., using James equation or related equations.

The effects of refraction on an X-ray entering the sample material aregenerally described with reference to FIG. 21.

Materials may have a negative index of refraction for X-rays. Asillustrated in FIG. 21, for example, at shallow angles, as X-rays from asource travels from a first medium, e.g., air to a second medium, e.g.,the material, where the mediums have different refractive indices, e.g.,n₁ and n₂, there is a change or shift in the propagation direction ofthe X-ray beam, illustrated by the solid lines. For example, as seen inFIG. 21, rather than proceeding through the second medium at theoriginal angle of the X-ray from the source, θ_(s), as depicted indashed and dotted lines, once the X-ray enters the second medium, theX-ray is instead refracted to an angle θ′_(s). Likewise, when the X-rayreflects off of the atomic plane of the second medium, the X-rayreflects within the second medium at an angle θ′_(d). As the X-rayreflection exits the second medium and enters the first medium, theX-ray reflection is further refracted from angle θ′_(d) to a finaldiffracted angle, illustrated by the solid arrow exiting the secondmedium.

Absent refractions due to different medium properties, i.e., if thefirst and second mediums have the same refractive index, the X-ray wouldtravel through the first and second mediums at the angle θ_(s), reflectfrom the atomic plane of the second medium at a reflection angle θ_(d),and travel from the second medium back to the first medium at the sameangle θ_(d).

As an example, the shift or change due to a difference in refractiveindexes of media may be similar to how an object appears to shift whenunder water as viewed from outside of the water. For example, a coin atthe bottom of a swimming pool filled with water may appear to be locatedin one location as observed from outside of the swimming pool butinstead may actually be located in a different location within the poolas confirmed by observing the coin from within the water of the pool.

Refraction effects at shallow incidence angles causes shifts in recordedTwo Theta peak positions of the sample, during measurements carried outusing high resolution grazing incidence X-ray diffraction stresstechnique. The residual stresses present in the sample may also cause ashift in the diffracted peak position. These competing effects may beaddressed during analysis of the measurement data, for example, byremoving or correcting for any Two Theta peak shifts caused byrefraction of X-rays at shallow incident angles. This allows the shiftsin diffracted peak position caused by residual stresses to be isolatedfrom refraction effects.

In some aspects, the effects of refractive index peak shifts may bedetermined according to the James equation (1963):

$\begin{matrix}{{2\; \theta_{B}} = {{2\; \theta_{L}} + {\frac{\delta}{\sin \mspace{14mu} 2\; \theta_{L}}\left\{ {2 + \frac{\sin ({b\prime})}{\sin \left( {{2\; \theta_{L}} + {b\prime}} \right)} + \frac{\sin \left( {{2\; \theta_{L}} + {b\prime}} \right)}{\sin ({b\prime})}} \right\}}}} & (5)\end{matrix}$

Where:

2θ_(B) is the measured Bragg angle

2θ_(L) is the true Bragg angle

b′ is the angle between the incident X-ray beam and sample surface

δ is the susceptibility of the material, as described earlier usingequation 2

As described in equation 5 above, the refractive index induced peakshift is affected by the incident angle of the X-ray beam, Two Thetaangle, material properties of the medium such as mass density and atomicnumber, the wavelength of the X-rays, real part of the dispersioncorrection of scattering factor, and other similar properties.

In some aspects, other formalisms related to or derived from Equation 5may also be used for calculating refraction induced peak shift.

The peak shift due to refraction effects is dependent on the Materialunder analysis (affected by atomic weight, density, atomic number, realpart of dispersion etc.), the wavelength of X-ray used, the grazingincident angle and Two Theta positions. As an example, in the case of agrazing incident diffraction measurement (0.84 degrees incident angle)on a Tungsten (W) film using Chromium K Alpha 1 X-rays, the W (110)reflection would appear at a Two Theta of 62.2 degrees instead of 61.78degrees, a 0.42 degree shift. In terms of d-spacing this corresponds toa shift of 0.0136 Angstroms. This shift needs to be accounted for inorder to carry out precise estimation of residual stresses present inthe Tungsten film or else inaccurate residual stress values may result.In addition this correction of the shift may be useful for keeping ad_(spacing) vs. cos² alpha*sin² ψ plot consistent for stressmeasurements carried out at different grazing incident angles on thesame sample.

Once the measurement data has been corrected for the Two Theta shiftcaused by the refraction of X-rays, the corrected measurement data maybe analyzed to determine whether any residual stresses are present.

Calculate Residual Stress

At 2008,

$\frac{d_{\psi} - d_{0}}{d_{0}}$

is plotted vs. cos² alphesin² ψ for each incident angle used during themeasurements as illustrated, for example, in FIG. 22. The d_(spacings)at different ψ positions, i.e., d_(ψ), may be calculated based on thecorrected Two Theta values, using Bragg's law as follows:

$\begin{matrix}{\lambda = {2*{dspacing}*{Sin}\mspace{14mu} \left( \frac{{Corrected}\mspace{14mu} {Two}\mspace{14mu} {Theta}}{2} \right)}} & (6)\end{matrix}$

Where:

λ is wavelength of X-rays used;

At 2010, a slope of the plot

$\frac{d_{\psi} - d_{0}}{d_{0}}\mspace{14mu} {{vs}.\mspace{14mu} \cos^{2}}\mspace{14mu} {alpha}^{*}\sin^{2}\; \psi$

is determined using any conventional method. For example, as illustratedin FIG. 22, the slope of the plot is −0.0063. As can be seen, the slopeof the plot has a R² value of 0.9977 which indicates how close the slopefits to a trend line of the data which is an improvement over simplyapplying the sin² ψ stress technique to the same sample as illustratedin FIG. 23 with a R² value of 0.9787. In addition, it is clearly shownhere that the High Resolution Grazing Incidence stress technique has agood correlation to the stress measured using traditional sin² ψ method.This is made possible because of the comprehensive method describedherein, which includes methods for instrument alignment, sampleplacement, adjustment, Omega offset correction, Two Theta selection,Refraction correction on measured Two theta values or d-spacing, etc.

At 2012, the residual stress may be calculated based on the slope of theplot using Young's modulus and Poisson's ratio according to thefollowing equation:

$\begin{matrix}{\sigma_{\phi} = {\frac{E}{\left( {1 + v} \right)}({slope})}} & (7)\end{matrix}$

Where:

σ_(φ) stress at a certain phi or sample rotation position;

slope is obtained from plot of

$\frac{d_{\psi} - d_{0}}{d_{0}}\mspace{14mu} {{vs}.\mspace{14mu} \cos^{2}}\mspace{14mu} {alpha}^{*}\sin^{2}\; \Psi$

E is Young's Modulus;

v is Poisson's Ratio;

ψ is the tilt angle of the sample;

d_(ψ)is the d-spacing at a certain ψ;

d₀ is the stress free d-spacing;

$\begin{matrix}{{alpha} = {\frac{{refraction}\mspace{14mu} {corrected}\mspace{14mu} {two}\mspace{14mu} {theta}}{2} - {{incident}\mspace{14mu} {angle}\mspace{14mu} \left( {{Omega}\mspace{14mu} {offset}\mspace{14mu} {corrected}} \right)}}} & (8)\end{matrix}$

As illustrated in FIG. 22, for example, the residual stress iscalculated as −1534±37 Mpa using the High Resolution Grazing Incidencesin² ψ technique. In contrast, as illustrated in FIG. 23, the residualstress calculated from the sin² ψ stress technique results in acalculated as −1553±139 MPa. While the residual stress value isapproximately the same, −1534 vs. −1553, the error bar for the HighResolution Grazing Incidence sin² ψ technique is much smaller, ±37 Mpavs. ±139 MPa, and provides a more definite estimation of the residualstresses present in the bulk material or thin film material of thesample.

In some aspects, one may also account for refraction effects by simplyaltering the stress free d-spacing d₀, by introducing refractioncorrection mechanisms described in this work, and in this case themeasured two theta or d-spacing does not have to be corrected.

At 2014, steps 2006-2012 may be repeated for different incident anglesif the stress gradient of the material needs to be determined.

At 2016, the residual stress as a function of x-ray penetration depthmay be plotted and output for presentation to a user, e.g., via adisplay of a display device.

Automated Testing and Analysis

In some aspects, some or all of the above methods, determinations,alignments, adjustments, measurements, scans, calculations or otherfeatures may be performed by a computing device 2410 with reference nowto FIG. 24.

In some aspects an automated sample handler systems using devices suchas robotic arm, robotic loader, sample tray, sample loader, conveyerbelt etc., or using other means, may be incorporated or used to enableautomated high resolution grazing incidence residual stress measurementsand analysis.

Computing device 2410 includes at least one processor 2412, memory 2414,at least one network interface 2416, a display 2418, an input device2420, and may include any other features commonly found in a computingdevice. In some aspects, computing device 2410 may, for example, be acomputing device associated testing apparatus 500 (FIGS. 5 and 6). Insome aspects, computing device 2410 may include, for example, a personalcomputer, laptop, tablet, smart device, or any other similar computingdevice that may be used by a user of testing apparatus 500. In someaspects, computing device 2410 may be integrated as part of testingapparatus 500.

Processor 2412 may include, for example, a microcontroller, FieldProgrammable Gate Array (FPGAs), or any other processor that isconfigured to perform various operations. Processor 2412 may beconfigured to execute instructions as described below. Theseinstructions may be stored, for example, in memory 2414.

Memory 2414 may include, for example, non-transitory computer readablemedia in the form of volatile memory, such as random access memory (RAM)and/or cache memory or others. Memory 2414 may include, for example,other removable/non-removable, volatile/non-volatile storage media. Byway of non-limiting examples only, memory 2414 may include a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a portable compact disc read-only memory (CD-ROM), anoptical storage device, a magnetic storage device, or any suitablecombination of the foregoing. In some aspects, memory 2414 may beconfigured to store measurement data 2422, the determined Omega offset2424, or any other setting or configuration for testing apparatus 500.

Network interface 2416 is configured to transmit and receive data orinformation to and from testing apparatus 500 or any other computingdevice via wired or wireless connections. For example, network interface2416 may utilize wireless technologies and communication protocols suchas Bluetooth®, WWI (e.g., 802.11a/b/g/n), cellular networks (e.g., CDMA,GSM, M2M, and 3G/4G/4G LTE), near-field communications systems,satellite communications, via a local area network (LAN), via a widearea network (WAN), or any other form of communication that allowscomputing device 2410 to transmit or receive information to or fromtesting apparatus 500.

Display 2418 may include any display device that is configured todisplay information to a user of computing device 2410. For example, insome aspects, display 2418 may include a computer monitor, television,smart television, or other similar displays. In some aspects, display2418 may be integrated into or associated with computing device 2410,for example, as a display of a laptop, smart phone, smart watch, orother smart devices, as a virtual reality headset associated withcomputing device 2410, or any other mechanism for displaying informationto a user. In some aspects, display 2418 may include, for example, aliquid crystal display (LCD), an e-paper/e-ink display, an organic LED(OLED) display, or other similar display technologies. In some aspects,display 2418 may be touch-sensitive and may also function as an inputdevice 2420. In some aspects, for example, computing device 2410 may beconfigured to present a plot of d_(spacing) vs. cos² alpha*sin² ψ to auser of computing device 2410 via display 2418.

Input device 2420 may include, for example, a keyboard, a mouse, atouch-sensitive display 2418, a keypad, a microphone, or other similarinput devices or any other input devices that may be used alone ortogether to provide a user with the capability to interact withcomputing device 2410.

In some aspects, processor 2412 may execute one or more programs orinstructions stored in memory 2414 to perform any of the above mentionedfunctions including, for example, apparatus calibration, calibratingsample alignment, determining the Omega offset, repositioning thesample, setting source and detector angles, identifying diffractionpeaks, selecting diffraction peaks, activating scans of the sample,calculating susceptibility of the material, calculating the Two Thetashift, plotting d_(spacing) vs. cos² alpha*sin² ψ, determining a slopeof the plot, and calculating a residual stress from the plot. In someaspects, for example, processor 2412 may input or determine some or allof sample code/name, Omega Offset value, Omega or incident angle, orOmega angle with Offset value incorporated, psi (w) or sample tiltvalues, Sample phi/rotation position, Two theta values incident X-raybeam, density of the material, atomic number of the material, the realpart of dispersion correction of scattering factor for the material,wavelength of the X-rays, mass attenuation coefficient of X-rays, theatomic mass of the material, stress free d-spacing/spacings, stress freetwo theta position, stress constant, Young's Modulus, Poisson's ratio,and Miller indices. In some aspects, for example, processor 2412 mayperform any of these functions automatically. For example, processor2412 may adjust the positioning of the sample and movement of the samplerelative to the source and detector by controlling one or more actuators(not shown) of the testing apparatus 500, may receive measurement data2422 from the detector 504, may receive or actively query one or moresources (not shown) or databases, locally or on the Internet, for theinput values described above including known properties of the materialsof the sample, and may analyze the measurement data as described aboveto calculate the residual stress of the materials of the sample. In someaspects, some or all of these functions may also require additional userinput or adjustment.

In some aspects, processor 2412 may receive inputs such as, for example,Omega Offset value, Omega or incident angle, or Omega angle with Offsetvalue incorporated, psi (w) or sample tilt values, Sample phi/rotationposition, Two theta values incident X-ray beam, X-ray wavelength etc.directly from the user, e.g., via input device 2420 or from computer feddata files, e.g., stored in memory 2414 or received via networkinterface 2416.

In some aspects, Two Theta peak diffraction peak positions forrefraction correction may be obtained from experimental data either bymanual selection or using peak deconvolution using standard or user madeor combined deconvolution methods/algorithms, using other methods, orusing a combination of methods using peak fitting or peak deconvolutionsoftware.

FIG. 25 illustrates a schematic of an example computer or processingsystem that may implement any portion of computing device 2410, testingapparatus 500, systems, methods, and computer program products describedherein in one embodiment of the present disclosure. The computer systemis only one example of a suitable processing system and is not intendedto suggest any limitation as to the scope of use or functionality ofembodiments of the methodology described herein. The processing systemshown may be operational with numerous other general purpose or specialpurpose computing system environments or configurations. Examples ofwell-known computing systems, environments, and/or configurations thatmay be suitable for use with the processing system may include, but arenot limited to, personal computer systems, server computer systems, thinclients, thick clients, handheld or laptop devices, multiprocessorsystems, microprocessor-based systems, set top boxes, programmableconsumer electronics, network PCs, minicomputer systems, mainframecomputer systems, and distributed cloud computing environments thatinclude any of the above systems or devices, and the like.

The computer system may be described in the general context of computersystem executable instructions, such as program modules, being executedby a computer system. Generally, program modules may include routines,programs, objects, components, logic, data structures, and so on thatperform particular tasks or implement particular abstract data types.The computer system may be practiced in distributed cloud computingenvironments where tasks are performed by remote processing devices thatare linked through a communications network. In a distributed cloudcomputing environment, program modules may be located in both local andremote computer system storage media including memory storage devices.

The components of computer system may include, but are not limited to,one or more processors or processing units 12, a system memory 16, and abus 14 that couples various system components including system memory 16to processor 12. The processor 12 may include a software module 10 thatperforms the methods described herein. The module 10 may be programmedinto the integrated circuits of the processor 12, or loaded from memory16, storage device 18, or network 24 or combinations thereof.

Bus 14 may represent one or more of any of several types of busstructures, including a memory bus or memory controller, a peripheralbus, an accelerated graphics port, and a processor or local bus usingany of a variety of bus architectures. By way of example, and notlimitation, such architectures include Industry Standard Architecture(ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA)bus, Video Electronics Standards Association (VESA) local bus, andPeripheral Component Interconnects (PCI) bus.

Computer system may include a variety of computer system readable media.Such media may be any available media that is accessible by computersystem, and it may include both volatile and non-volatile media,removable and non-removable media.

System memory 16 can include computer system readable media in the formof volatile memory, such as random access memory (RAM) and/or cachememory or others. Computer system may further include otherremovable/non-removable, volatile/non-volatile computer system storagemedia. By way of example only, storage system 18 can be provided forreading from and writing to a non-removable, non-volatile magnetic media(e.g., a “hard drive”). Although not shown, a magnetic disk drive forreading from and writing to a removable, non-volatile magnetic disk(e.g., a “floppy disk”), and an optical disk drive for reading from orwriting to a removable, non-volatile optical disk such as a CD-ROM,DVD-ROM or other optical media can be provided. In such instances, eachcan be connected to bus 14 by one or more data media interfaces.

Computer system may also communicate with one or more external devices26 such as a keyboard, a pointing device, a display 28, etc.; one ormore devices that enable a user to interact with computer system; and/orany devices (e.g., network card, modem, etc.) that enable computersystem to communicate with one or more other computing devices. Suchcommunication can occur via Input/Output (I/O) interfaces 20.

Still yet, computer system can communicate with one or more networks 24such as a local area network (LAN), a general wide area network (WAN),and/or a public network (e.g., the Internet) via network adapter 22. Asdepicted, network adapter 22 communicates with the other components ofcomputer system via bus 14. It should be understood that although notshown, other hardware and/or software components could be used inconjunction with computer system. Examples include, but are not limitedto: microcode, device drivers, redundant processing units, external diskdrive arrays, RAID systems, tape drives, and data archival storagesystems, etc.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, the cloud, a local areanetwork, a wide area network and/or a wireless network. The network maycomprise copper transmission cables, optical transmission fibers,wireless transmission, routers, firewalls, switches, gateway computersand/or edge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

Although specific embodiments of the present invention have beendescribed, it will be understood by those of skill in the art that thereare other embodiments that are equivalent to the described embodiments.Accordingly, it is to be understood that the invention is not to belimited by the specific illustrated embodiments, but only by the scopeof the appended claims.

1. A method comprising: performing an Omega scan to determine an Omegaoffset of an X-ray beam generated by an X-ray source relative to amaterial with respect to an incidence angle between the X-ray source andthe material; setting the incidence angle between the X-ray source andthe material at a first target incidence angle, the first targetincidence angle based on the Omega offset and greater than the criticalangle of the material; performing a grazing incidence X-ray diffractionscan on the material, the grazing incidence X-ray diffraction scangenerating first measurement data comprising intensities of X-rayphotons at a plurality of two theta angles; analyzing the firstmeasurement data to identify a plurality of diffraction peaks from thematerial, each diffraction peak having an intensity occurring at acorresponding two theta value; selecting a diffraction peak of theplurality of diffraction peaks based on the analysis of the firstmeasurement data; setting the incidence angle between the X-ray sourceand the material at a second target incidence angle based on the Omegaoffset and a desired penetration depth into the material; performing twotheta scanning on the material on a range of two theta values around thetwo theta value of the selected diffraction peak at a plurality of tiltpositions, the two theta scanning generating second measurement data;applying refraction correction to the second measurement data, therefraction correction correcting the second measurement data for eachtilt position of the scanned range of two theta values around the twotheta value of the selected diffraction peak converting the correctedsecond measurement data measured at each tilt position to a d-spacingfor each tilt position; and determining residual stress values of thematerial based on the converted corrected second measurement data. 2.The method of claim 1, wherein the material is selected from the groupconsisting of a thin film material on a substrate and a bulk material.3. The method of claim 1, further comprising determining thesusceptibility of the material, wherein the critical angle of thematerial is calculated based on the determined susceptibility.
 4. Themethod of claim 3, wherein applying the refraction correction furthercomprises calculating the two theta shift due to the refraction ofX-rays in the material based on the determined susceptibility and Jamesequation.
 5. The method of claim 1, wherein for the two theta scanningcarried out at the plurality of tilt positions, the range of two thetascans may be selected based on a proximity to the selected diffractionpeak.
 6. The method of claim 1, further comprising comparing the twotheta values of the plurality of diffraction peaks to determine adistance in two theta between each adjacent diffraction peak, whereinselecting the diffraction peak from the plurality of diffraction peakscomprises selecting the diffraction peak based on the determineddistance in two theta between each adjacent diffraction peak.
 7. Themethod of claim 6, further comprising determining which diffraction peakof the plurality of diffraction peaks has a greater distance in twotheta from adjacent peaks, wherein selecting the diffraction peakcomprises selecting the diffraction peak that has a greater distance intwo theta from adjacent peaks.
 8. The method of claim 1, furthercomprising: setting the incidence angle between the X-ray source and thematerial at a third target incidence angle based on the Omega offset anda second desired penetration depth into the material; and performing aseries of two theta scans on a range of two theta values around the twotheta value of the selected diffraction peak at a plurality of tiltpositions, the series of two theta scans generating third measurementdata, wherein the residual stress values for the material are determinedbased at least in part on the third measurement data.
 9. The method ofclaim 1, further comprising plotting d_(spacing) vs. cos² alpha*sin² ψfor a plurality of penetration depths on the material.
 10. The method ofclaim 1, wherein the method is performed automatically by at least oneprocessor comprising hardware.
 11. A computer readable medium comprisinginstructions that, when executed by at least one processor comprisinghardware, configure the at least one processor to: perform an Omega scanto determine an Omega offset of an X-ray beam generated by an X-raysource relative to a material with respect to an incidence angle betweenthe X-ray source and the material; set the incidence angle between theX-ray source and the material at a first target incidence angle, thefirst target incidence angle based on the Omega offset and greater thanthe critical angle of the material; perform a grazing incidence X-raydiffraction scan on the material, the grazing incidence X-raydiffraction scan generating first measurement data comprisingintensities of X-ray photons at a plurality of two theta angles; analyzethe first measurement data to identify a plurality of diffraction peaksfrom the material, each diffraction peak having an intensity occurringat a corresponding two theta value; select a diffraction peak of theplurality of diffraction peaks based on the analysis of the firstmeasurement data; set the incidence angle between the X-ray source andthe material at a second target incidence angle based on the Omegaoffset and a desired penetration depth into the material; perform twotheta scanning on the material on a range of two theta values around thetwo theta value of the selected diffraction peak at a plurality of tiltpositions, the two theta scanning generating second measurement data;apply refraction correction to the second measurement data, therefraction correction correcting the second measurement data for eachtilt position of the scanned range of two theta values around the twotheta value of the selected diffraction peak; convert the correctedsecond measurement data measured at each tilt position to a d-spacingfor each tilt position; and determine residual stress values of thematerial based on the converted corrected second measurement data. 12.The computer readable medium of claim 11, wherein the material isselected from the group consisting of a thin film material on asubstrate and a bulk material.
 13. The computer readable medium of claim11, the instructions further configuring the at least one processor todetermine the susceptibility of the material, wherein the critical angleof the material is calculated based on the determined susceptibility.14. The computer readable medium of claim 13, wherein applying therefraction correction further comprises calculating the two theta shiftdue to the refraction of X-rays in the material based on the determinedsusceptibility and James equation.
 15. The computer readable medium ofclaim 11, wherein for the two theta scanning carried out at theplurality of tilt positions, the range of two theta scans may beselected based on a proximity to the selected diffraction peak.
 16. Thecomputer readable medium of claim 11, the instructions furtherconfiguring the at least one processor to compare the two theta valuesof the plurality of diffraction peaks to determine a distance in twotheta between each adjacent diffraction peak, wherein selecting thediffraction peak from the plurality of diffraction peaks comprisesselecting the diffraction peak based on the determined distance in twotheta between each adjacent diffraction peak.
 17. The computer readablemedium of claim 16, the instructions further configuring the at leastone processor to determine which diffraction peak of the plurality ofdiffraction peaks has a greater distance in two theta from adjacentpeaks, wherein selecting the diffraction peak comprises selecting thediffraction peak that has a greater distance in two theta from adjacentpeaks.
 18. The computer readable medium of claim 11, the instructionsfurther configuring the at least one processor to: set the incidenceangle between the X-ray source and the material at a third targetincidence angle based on the Omega offset and a second desiredpenetration depth into the material; and perform a series of two thetascans on a range of two theta values around the two theta value of theselected diffraction peak at a plurality of tilt positions, the seriesof two theta scans generating third measurement data, wherein theresidual stress values for the material of the material are determinedbased at least in part on the third measurement data.
 19. The computerreadable medium of claim 11, the instructions further configuring the atleast one processor to plot d_(spacing) vs. cos² alpha*sin² ψ for aplurality of penetration depths on the material.
 20. (canceled)